Differential Equations

posted by .

Model radioactive decay using the notation
t = time (independent variable)
r(t) = amount of particular radioactive isotope present at time t (dependent variable)
-λ = decay rate (parameter)
Note that the minus sign is used so that λ > 0
a) Using this notation, write a model for the decay of a particular radioactive isotope.
b) If the amount of the isotope present at t = 0 is r0, state the corresponding initial-value problem for the model in part (a).

  • Differential Equations -

    the rate is proportional to r(t) times a constant

    dr/dt=r(t)* constant where the constant is negative

    and the solution to this first order diff equation is of the form

    r(t)=K*e^(- λ)t + C

    b. ro=K+C
    and at t=inf, r(inf)=0
    which implies C is zero, so k=ro.
    r(t)=ro*e^- λ t

  • Differential Equations -

    a) dr(t)/dt = -λr(t)

    =>∫[ dr(t)/r(t) ] = ∫[ -λt ]
    => ln |r| = -λt + C
    => r(t) = e^(-λt + C) = e ^ (C - λt)
    => r(t) = e^C * e ^ (-λt)

    let C = e^C
    => r(t) = Ce^(-λt)

    b) r(0) = r_0

    =>r(0) = r_0 = Ce^(-λ(0))
    =>r_0 = C

    =>r(t) = r_0 * e^(-λt)

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    Carbon-14 is a radioactive substance produced in the Earth's atmosphere and then absorbed by plants and animals on the surface of the earth. It has a half-life (the time it takes for half the amount of a sample to decay) of approximately …
  2. Math - D.E.Q.

    The half-life of a radioactive isotope is the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount. i) The half-life of Carbon 14 (C-14) is 5230 years. Determine the decay-rate …
  3. ap calculus

    suppose that the amount in grams of a radioactive substance present at time t (in years) is given by A (t) = 160e-.70t. Find the rate of decay of the quantity present at the time when t = 4
  4. Math

    A radioactive substance decays according to the formula Q(t) = Q0e−kt where Q(t) denotes the amount of the substance present at time t (measured in years), Q0 denotes the amount of the substance present initially, and k (a positive …
  5. Math

    A radioactive substance decays according to the formula Q(t) = Q0e−kt where Q(t) denotes the amount of the substance present at time t (measured in years), Q0 denotes the amount of the substance present initially, and k (a positive …
  6. Calculus

    My teacher wants me to use this formula for half life question C(t) = Ce^-kt but i used a different formula plz help. the question is: A radioactive isotope has a half life of 30 years. If we started off with 10 mg of this isotope …
  7. chem

    a) Potassium-40 undergoes three different modes of radioactive decay (Beta decay, Positron decay, and Electron Capture decay). Write balanced nuclear reactions for each of these decay modes. Are the products stable (not radioactive)?
  8. chem

    Potassium-40 undergoes three different modes of radioactive decay (Beta decay, Positron decay, and Electron Capture decay). Write balanced nuclear reactions for each of these decay modes. Are the products stable (not radioactive)?
  9. physics

    a radioactive sample has a decay rate R of 518 decay/min at time t=7 min and 156 decay/ min at time t=17 min. calculate the decay constant and the initial decay rate?
  10. Calculus

    The rate of decay is proportional to the mass for radioactive material. For a certain radioactive isotope, this rate of decay is given by the differential equation dm/dt = -.022m, where m is the mass of the isotope in mg and t is the …

More Similar Questions